The Kinetics of HIV Infectivity - Federation of American ... · The Kinetics of HIV Infectivity The virologist would set about answering our questions by running a series of viral

The Kinetics ofHIV Infectivity

by Scott P. Layne, Micah Dembo,and John L. Spouge

s uppose that we give twovirologist identical samples ofhuman immunodeficiency virus(HIV) and ask each of them todetermine some simple proper-ties. Questions that we might askinclude: How many infectiousvirus particles, or virions, are inthe sample? How virulent are thevirions? How stable are the viri-ons? How effective are variouschemical agents in blocking infec-tion? How effective are antibod-ies from infected individuals inneutralizing the virions?

91

The Kinetics of HIV Infectivity

The virologist would set about answering our questions by running a seriesof viral infectivity assays, in which specific conditions would be tailored to tackleeach particular problem (Fig. 1). For example, to deal with the question of blockingagents, the virologist would inoculate aliquots of our sample virus into a series ofchambers, each containing target cells plus a different concentration of blocker. Theeffectiveness of the blocker could be judged by the amount required to reduce target-cell infection by one-half relative to an untreated control. Despite the superficial ap-pearance of scientific rigor, it would not be surprising to find that the two virologist(with the best of intentions and technique) obtained significantly different results forthe same blocking agent and viral strain. With further inquiry we would most likelydiscover that the virologist used somewhat different assay conditions, that is, differ-ent target-cell types, cell concentrations, viral-inoculation techniques, and so on. Tounderstand the underlying causes of the discrepancy, we would need to integrate agreat deal of detailed and quantitative information. Only then could we judge whichresult might be most representative of the agent’s activity in clinical situations.

The difficulty of comparing the results of one assay method with those of an-other is one of the biggest headaches in virology. That problem is particularly im-portant in the case of HIV because the screening for potential therapeutic agents andvaccines against HIV is frequently based on assay results alone. To improve the util-ity of such assays, it is useful to study theoretical models of the kinetic processes thatdetermine their outcomes.

In this article we present a model for the kinetics of HIV infection in assay sys-tems. We show how to use the model for designing and analyzing viral infectivityassays and for answering the kinds of questions posed above. We also use the modelto evaluate prospects for blocking therapies and vaccines.

Elements of a Viral Infectivity Assay

HIV infects subsets of lymphocytes, monocytes, and macrophages exhibitingthe CD4 protein on their surfaces. Such CD4+ cells manifest infection through aspectrum of outcomes ranging from prolonged latency to cell fusion and cell death.Sometimes replication is so explosive that target cell membranes lyse as newbornvirions emerge. Unfortunately, CD4+ cells are at the helm of the immune system’sresponse to microbial invasion. Thus HIV infection not only harms individual tar-get cells but also perturbs the entire communication network for the body’s defenses.The result is a catastrophic susceptibility to opportunistic infections. (For more de-tails see “AIDS Viruses in Animals and Man: Nonliving Parasites of the ImmuneSystem.”)

Viral infectivity assays involve great kinetic complexity, but they are still muchsimpler than the infective process in vivo. For example, in assays only a single typeof target cell is present and only the primary infections caused by the initial virionsadded to the assay chamber are studied (Fig. 1). In vivo, direct cell-to-cell transmis-sion of infection and the prolonged growth and reproduction of the virus come intoplay. Assays also neglect the effects of the normal immune responses blocking infec-tion; clearly these are extremely important in vivo.

To formulate a model of an assay system, lets start by considering an ideal-

Los Allamos Science Fall 1989


Preincubation PhaseI

Birth of Viral CohortShedding of gp120Nonspecific Killing of Virus

Primary Infection Phase

Addition of Virusto Reaction Chamber

Initial Infectionof Target Cells

Beginning of ViralReplication

Timev

Secondary InfectionPhase

Budding of New Virionsfrom Infected Cells

Infection of Target Cellsby New Virions

PHASES IN A VIRAL INFECTIVITYASSAY

Fig. 1. Schematic diagram of the three phasesof a viral infectivity assay. The inoculum isprepared by growing HIV in cell cultures andcentrifuging the supernatant to separate viri-ons from cellular debris and is then stored for

a “pre-incubation” time Tp. For simplicity Weassume that the inoculum consists of a “ho-mogeneous cohort” of virions born at T = – Tp.

0, the virions become less infectious as theyshed gp120 and can also be killed by non-specific agents, but target-cell infection doesnot occur. To begin the primary infectionphase, a calibrated number of virions is addedto the target cells in the reaction chamber atT = O. During the primary infection phase,

specific killing, and target-cell infection occur.Ordinarily HIV replicates within 24-48 hours

During the secondary infection phase, T >105 seconds, new virions emerge from cells in-fected during the primary infection phase andsecondary infections occur. The secondarilyinfected cells then produce new virions, whichinfect other cells, and so on. A viral infectivityassay is said to be linear if the additional cy-cles of infection produce virions (or viral pro-teins) in proportion to the number of initial in-fections.

Los Alamos Science Fall 1989 93


RELATING THEORY TO EXPERIMENT

Fig. 2. Most experimental assays determine I,the number of cells produced during the pri-mary infection phase (see Fig. 1) by a dilutionmethod known as the ID-50 method. As shownon the diagram, a viral stock solution is inoc-ulated into a large number of assay chambers(usually 10 to 20) and the infection is allowedto go to completion. The experiment is thenrepeated again and again, each time with amore dilute solution, until no infections occurin half the chambers. The reciprocal of thedilution factor required to achieve this resultis called ID-50 (Infectious dose-50 per cent).Now, when half the chambers have no infec-tion, we can say that the probability of the di-luted viral stock solution’s infecting O cells is

a viral stock solution’s infecting k cells is de-scribed by a Poisson distribution, that is, since

Thus I diluted = In 2. We determine I for theoriginal stock solution by multiplying In 2 byID-50. Having determined I, it can be used to

the assay is performed under conditions such

ized solution containing a “homogeneous cohort” of free virions of a particular HIVstrain. A homogeneous cohort is defined as a population of virions that were bornsimultaneously and that have been treated identically ever since. (An actual stock so-lution of virus will consist of a mixture of homogeneous cohorts but it is simplest totreat each cohort separately. This involves no loss of generality, since we can readilyobtain all the properties of a mixture of cohorts by taking a weighted average.) Wecan imagine that the members of the homogeneous cohort were born at some time inthe past T = – TP.

At T = O, we inoculate V0 random members of the viral cohort into a chambercontaining a much larger number of CD4+ target cells. At some later time T > 0,some of these will have successfully infected target cells. That number, call it I(T),is the primary quantity of biological interest. I(T) is related to the probability that asingle virion will successfully infect a target cell by time T; that is, i = I/VO,. It isworth mentioning that if we keep the size of our inoculum sufficiently small relativeto the number of target cells then the number of successfully infecting virions will beequal to the number of infected cells. Figure 2 describes how I is actually measuredexperimentally.

Once an assay is underway, a number of kinetic processes occur simultaneously.Infection of target cells occurs in multiple stages: first, a virus diffuses to the cellsurface; second, the gp120 glycoproteins on the virus’s surface and CD4 on the tar-get cell’s surface form bimolecular complexes; and third, interactions involving CD4,gp120, and gp41 (which is attached to gp120) promote fusion of the HIV envelopewith the target-cell membrane, resulting in entry of the viral core into the cell. Sub-sequently reverse transcription of the viral genome, its incorporation into the genomeof the host, and production of new virions complete the life cycle. Before they pen-etrate target cells, the virions in the assay chamber are subject to several degrada-

94 Los Alamos Science Fall 1989


INDIVIDUAL REACTIONS IN THE KINETIC MODEL

Infection

HIV

Membrane

Fig. 3. Schematic diagram of the kinetic pro-cesses in a viral infectivity assay. The levelof detail shown here mirrors the treatment inour kinetic model. (a) The process that com-plicates the kinetics of the model is the spon-taneous shedding of gp120 from so-called livevirions; that is, gp120 spontaneously dissoci-ates from gp41. The rate constant for shed-ding of gp120 is ks. Once a gp120 molecule isshed, it cannot bind again to a virion so we donot keep track of it in our kinetic model. Theirreversible shedding process causes progres-sive inactivation of virions; that is, the viri-ons become less and less infectious. Whenthey lose all of their gp120s, they are inactivebut still subject to nonspecific killing mech-anisms, shown schematically in (b). Non-specific killing mechanisms include enzymaticdegradation and dissolution by soaps such asnonoxynol-9, a component of common spermi-cides. In (c) we show an infective event, thatis, the entry of a virion into a target cell whereit can begin to replicate. We do not modelthat process in detail but rather assume thetotal rate of infection in the assay chamber isproportional to keF, where F is the number of

rate constant that combines all quantities in-volved in collision of a virion and a target cell,binding of gp120 to CD4, fusing of the virionwith the cell membrane, entry of the viral core,and integration of the viral genome into thegenome of the target cell. We assume ke re-mains constant during the assay, whereas Fis continuously changing. In (d) we show thereversible blocking reaction between solubleCD4 (sCD4) and the gp120 molecules on thesurface of a live virion. The parameters kf andkr are the forward and reverse rate constants,respectively, for the completing of gp120 withsCD4 (that Is, for the formation and dissoci-ation of gp120-sCD4 complexes). Those pro-cesses result in the masking and unmaskingof gp120s but do not result in a net loss ofgp120 from the reaction chamber nor in thedisappearance of live virions.



TIME DEPENDENT KINETICS

Fig. 4. All processes shown In Fig. 3 occursimultaneously during the primary infectionphase of a viral infectivity assay. Notice that avirion may exist in one of many different statesat the time it encounters a target cell. A livevirion that has lost all its gp 120s cannot Infecttarget cells and is therefore inactive. As timeprogresses, the live virions continually shedgp120 from their surfaces. Also, the blockersCD4 may bind to and then dissociate from thefree gp120s on the virion surfaces. The rate ofInfection of target cells Is assumed to be pro-portional to keF. We assume that an infectiveevent causes the loss from the medium of theinfecting virion along with the free gp120s andgp120-sCD4 complexes on its surface.

tive processes that alter their ability to complete a life cycle: most important, HIVspontaneously sheds, over a period of hours, the seventy to eighty gp120 moleculespresent on its surface at birth. After shedding all its gp120s, a virion can no longerbind to a target cell. HIV may also be under attack by antibody, by enzymes natu-rally present in the surrounding medium, or by other viracidal agents added to theassay chamber. Neutralization, degradation, and dissolution of the virions by suchagents is usually irreversible and will be referred to as nonspecific killing. Finally,the infectiousness of virions may be reversibly inhibited by a blocking agent (forexample, soluble CD4, abbreviated sCD4) that forms a complex with gp120 andthereby prevents it from binding to target cells. Figure 3 illustrates schematically therate laws that we propose govern the kinetic processes described above. Figure 4 il-lustrates how the processes are integrated to give a closed system of equations. Weshould emphasize that, although these rate laws are plausible, definitive tests are notyet available.

The Rate Equations

The rate equations are formulated in terms of four dependent variables F,C,V,and I. These represent, respectively, the number of free gp120 molecules present

9 6 Los Alamos Science Fall 1989


on living virions in the assay chamber, the number of complexed gp120 moleculespresent on living virions in the assay chamber, the number of living viral particles inthe chamber, and the number of infected cells in the chamber. There are also severalindependent variables; these are: time T, the preincubation time Tp, the concentrationB of the blocking agent sCD4, the size of the inoculum Vo, and the concentrationof target cells L. Finally, there are constant parameters: the initial number of gp120molecules on each virion at birth N and the five rate constants kl, kn, ks, kf, and kr

defined in Fig. 3. In terms of these quantities the rate equations are:Rate of Target-Cell Infection:

Rate of Loss of Live Virions:

Rate of Change of Number of Free gp120s:

Rate of Change of gp120-sCD4 Complexes:

(1)

(2)

(3)

(4)

The term keLF on the right of Eq. 1 represents the total rate of target-cell in-fection in the assay chamber. This rate law incorporates the notion that infection re-quires the combination of a free gp120 molecule with a target cell. Moreover, thisrate law assumes that in some sense all gp120s are equivalent in their potential toinitiate an infection. One corollary of this assumption of equivalence is that an indi-vidual virus particle must infect at a rate proportional to the number of free gp120son its surface.

The term knV on the right of Eq. 2 represents the rate of loss of live virions dueto nonspecific killing. Note that virions are also lost at a rate keLF, due to penetra-tion of target cells.

The terms kfBF and krC on the right of Eqs. 3 and 4 are the rates of formationand dissociation of gp120-sCD4 complexes, respectively. The terms (ks + kn)F and(ks + kn)C are the rates of loss of free and complexed gp120, respectively, due to thecombined effects of spontaneous shedding and nonspecific killing. These rate lawsimply that when a virus is nonspecifically “killed” there is concomitant loss of all thefree and complexed, gp120s on its surface. As far as our theory is concerned, thesecompletely disappear from the system and have no further kinetic role. In a similarway, shedding of a free or complexed gp120 molecule is equivalent to loss of thismolecule from the system. It is also apparent that for these rate laws to hold exactly,

Los Alamos Science Fall 1989 9 7


the processes of nonspecific killing and shedding must operate equally on all the freeand complexed gp120 molecules in the system.

rates of loss of free and complexed gp120 molecules due to infective events. Onceagain these terms hold rigorously only if each free gp120 remaining in the systemhas an equal chance of initiating an infective event. The proof proceeds from thefact that only one free gp120 is required to initiate an infective event. There willalso be some additional gp120s that disappear (are internalized) as a byproduct ofan infective event. By calculating the appropriate conditional expectations, one canshow that the average number of such losses per event will be (N – l)F/NV free and(N – 1)C/NV complexed gp120 molecules.

To uniquely solve the rate equations, we must specify the conditions at the startof the primary infection phase (T = O). To do this in a simple way, let’s supposethat before the virions are added to the assay chamber, they are isolated from targetcells and blocker. Therefore, L, B, I, and C are all equal to zero between T = –Tp

and T = O. Solving Eqs. 2–5 we easily find that the initial conditions at T = O are :I = O, V = Vo exp (–k nTP), F = NVoexp [–(kn + ks)T p] , and C = O.

Parameter Estimation

By proposing our model we have, at least in principal, reduced the problem ofphysically characterizing the kinetics of viral infection to the problem of determin-ing N and the five rate constants ke, kf, kr, ks, and kn. Unfortunately, satisfactoryexperimental measurements of these parameters are not yet available. Thus the prob-lem of designing experiments for determining parameters is one of the major goalsof our analysis (see following section on determining rate constants). For the presentwe will simply indicate how one may derive some rough a priori estimates of theparameters. These will be useful for generating illustrative numerical solutions (seefollowing section).

The rate constant for target cell infection kt, is an overall rate constant for themultiple stages involved in the infection of a target cell by a virion (that is, its bind-ing to an sCD4 receptor, its fusing with the cell, and so on). It should be remem-bered however that the rate constant is defined on a “per free gp120” basis and noton a “per virus” basis. From this definition it can easily be seen that the upper limitof ke must occur when every diffusive encounter between a free gp120 molecule at-tached to a viron and a target cell results in infection. After taking into account thediffusion constants of a virion and of a target cell and after correcting for the frac-tion of the virion’s surface that is covered by a single gp120, the diffusion-limitedapproximation yields a value for k l of about 8 x 10-13 c m3 S –l.

Similarly, the diffusion-limited approximation for the rate constant for colli-sion between sCD4 and gp120 yields an upper limit for the forward rate constant for

To estimate a value for kr, the reverse rate constant for blocking, we use the re-sults of equilibrium-binding experiments. Those tell us that Kassoc, the equilibriumconstant for association between sCD4 and gp120 is about 2.0 x 10–12 cm3 permolecule. Then, since K assoc = k f/ kr, we estimate that k r = kf/ Kassoc < 1.5 s-1.

98


Electron-microscope studies on the structure of HIV indicate that seventy toeighty gp120 molecules completely cover the surface of a mature virion. Therefore,we take N = 80.

Finally, to estimate the rate constant for nonspecific killing kn and the rate con-stant for shedding of gp120 from a live virion ks, we use Peter Nara’s results for twostrains of the human T-lymphotropic virus III. He showed that those strains lose halfof their infective activity within 4 to 6 hours when incubated in their growth media at37°C. That implies that both ks and kn are less then about 10-4 S -l.

Numerical Solutions

Having specified the inital conditions and obtained estimates for the rate con-stants, we can solve the rate equations numerically. The sample solutions shownin Fig. 5 help us gain insight into the temporal behavior of the model. The aver-age numbers of live virions, infected target cells, and free gp120’s and gp120-sCD4complexes on the surface of live vinons are desplayed as nondimensional variables(see “Mathematical Considerations” for definitions). Figure 5a shows a case with noblocker (B = O); Fig. 5b and Fig. 5C show the effect of adding a low and a higherconcentration of blocker; Fig. 5d shows the effects of a high concentration of blockerin conjunction with a nonspecific killing agent (for example, nonoxynol-9).

When no blocker is present (Fig. 5a), the number of infected target cells rises

target-cell infection diminishes. The obvious decline in the number of live virions

target cells; the remaining 28 per cent, now completely lacking gp120 molecules andhence inactive or noninfectious, remain in the medium. Since nonspecific killing wasnot included in this computation, the inactive live viral particles remain in solutionindefinitely.

Figure 5b shows the effects of adding a low concentration of the blocking agentsCD4 to the culture medium. The presence of sCD4 does not affect the rate of target-

After that time the rate of target-cell infection declines by a factor of 2 to 3,and ultimately, only 35 per cent of the live virions infect target cells. The declinecompared to case (a), 35/72, is approximately the proportion of gp120s blocked bySCD4.

Figure 5C shows the effects of a hundredfold higher concentration of sCD4.Equilibration between the blocker and gp120 occurs more rapidly, in only 3 x 10-3

second, and 199 out of every 200 gp120 molecules are blocked. The percentage oflive virions that infect target cells declines from 72 per cent in case (a) to only 0.36per cent.

Figure 5d shows the synergy of sCD4 and a nonspecific killing agent. We assigna rate constant for nonspecific killing, kn = 5 x 10–4 S–l, that is fivefold greater thanthe rate constant assumed for gp120 shedding, ks = 10–4 S–l. As in Fig. 4c, bindingof sCD4 to viral gp120 and shedding of viral gp120 are assumed to be independent



TIME DEPENDENT SOLUTIONSOF THE KINETIC MODEL

Fig. 5. Four numerical solutions of the modelillustrating the progress of an untreated infec-tion, an infection treated with two concentra-tions of sCD4, and an infection treated withsCD4 plus nonspecific killing agent. The cor-responding parameters are (a) B = O andkn = O, (b) B = 1 012 molecules cm–3 a n dkn = O, (c) Bkn = O, and (d) B = 1014 molecules cm–3 andkn = 5 X 10- 4 s -1 . The results are plottedin non-dimensional variables: “Infected Cells”

Tp = O and L = 2 x 106 cells cm–3, whichis a typical lymphocyte concentration for in-fectivity assays.

100

Time

of nonspecific killing, which occurs on a “per live virion” basis. Nonspecific killingcauses the disappearance of virions and so infection stops before virions shed all oftheir gp120 molecules. As a result, the ultimate number of infective events per virionis sixfold less than the case shown in Fig. 5c.

Although the time development of infection is of interest, it is usually not pos-sible to freeze the infective process at an intermediate stage. Thus, most experiments

Los Alamos Science Fall 1989


101


NONLINEAR EFFECTS OFTARGET-CELL CONCENTRATION

Fig. 6. Numerical solutions of the model il-

(see Eqs. 8’ and 9’ in “Mathematical Consider-ations.”) The four solutions correspond to dif-ferent sCD4 concentrations: B = 0, 1012, 1013

–3and 1014 molecules cm . The life of a virionconsists of a race between finding a target ceil

uation in which a virion is likely to have onlyone chance to infect a target cell in its lifetime.

ation in which a virion has multiple chancesto infect a target cell. Since we assumedin these calculations that the pre-incubation

breakdown of this proportionality occurs as

also shows the effects of adding various con-centrations of sCD4. Notice that the region oftransition between linear and nonlinear behav-ior depends strongly on the blocker concentra-tion. For all solutions kn = O, k s = 10 –4 S –l,and the primary infection time is 18 hours, or6.48 x 104 seconds.

1.0

0.8

0.2

0.0

< + N - l

B = 10 ‘4

0.0 2.0 4.0 6.0 8.0 10.0

Target-Cell Concentration, L (106/cm3)

viral infectivity assays. It says that for some purposes, such as measuring blocker

different assays much more difficult.

Determining Rate Constants from Experiment

A primary motivation for developing a kinetic theory is to provide a means fordefining and determining meaningful parameters. For the case at hand the main un-

we are also ignorant of the exact size of the initial inoculum Vo. Our analysis of themodel (see “Mathematical Considerations”) indicates that a completely satisfactorysolution to the problem of parameter determination is not really possible. For exam-ple, it is very difficult to determine the values of the rate constants kf, and kr sepa-rately. One must thus be satisfied with simply determining the ratio of these quanti-ties K assoc = k f/ kr.

Accurate determinations of Kassoc are needed to assess the effectiveness of agentssuch as sCD4 in blocking infection. To design an experiment to determine Kassoc, we



2,0

1.0

I

L = 2 X 104

\

the blocker concentration B, and all other variables would be held constant. For con-

the slope of such a plot will be a good estimate of Kassoc. To simulate such an ex-periment, we generated numerical solutions to Eqs. 2–5 for various values of B andplotted the appropriate ratio versus B (Fig. 7). Although the plots for all values of Lappear linear, only at the lowest target-cell concentrations is the slope a reasonableestimate of Kassoc.

A number of publications report that sCD4 blocks HIV infection of CD4+ lym-phocytes, but only two provide sufficient information for determining Kassoc with thetechnique shown in Fig. 7. From published data of Deen et al., we determine thatK

3.8 x 10–12 cm3 molecule-l for two sCD4 derivatives.Those “biological” results should be compared to the following range of val-

2.3 x 10-12 cm3 molecule -1 for various analogs of sCD4. The close agreement be-tween biological and physical methods strongly supports the fundamental assumptionof our model that infection proceeds at a rate proportional to the number of free gp-120s per virion (equivalent-site approximation). That agreement would not ensue ifthere were significant infection mechanisms not requiring gp120 nor if blocking es-sentially all gp120s was necessary to diminish infection. Since blocking is reversible,it would not ultimately block infection in the absence of nonspecific killing and shed-

MEASURING BLOCKER AFFINITY

Fig. 7. Numerical solutions simulating a seriesof infectivity assays for quantifying blockeraffinity (Kassoc). Affinity is measured by com-paring results of an assay without blockerwith those of an assay with blocker, holdingother conditions constant. This “control’’-to-

spond to increasing concentrations of targetcells: L = 2 x 104, 2 x 105, 2 x 106, 2 X 107,and 2 x 108 cells cm ‘3 The slopes corre-

sponding to those concentrations are 2.0 x10 -12, 1.9 x 10-12, 1.1 x 10-12, 8.0 x 10-14,and O cm3 molecule–l, respectively. Accord-ing to Eq. 8’, the slopes provide estimates of

the “apparent” equilibrium constant for the as-sociation of blocker and gp120. The slope ofthe curve corresponding to the lowest valueof L (the top curve) yields the best estimateof K assoc. For ail solutions T P = O, k n =

o , k s = 1 0 -4 s - 1,at T = 6.48 x 104 seconds.



MEASURING THE NONSPECIFICKILLING AND SHEDDINGRATE CONSTANTS

Fig. 8. Five numerical solutions of the modelsimulating a series of experiments to deter-mine ks and kn. In each simulation the viri-ons pre-incubate for various times TP beforebeing added to a reaction chamber. The fiveplots correspond to increasing concentrationsof nonspecific killing agent: kn = O, 0.5 x1 0–4, 10–4, 2 x 10–4, and 4 x 10–4 s –l. Theordinates are normalized by the initial num-ber of virions, a procedure equivalent to tak-ing V o = 1 in Eqs. 8’ and 9’. Initially theslope of each plot is kn, but at longer pre-incubation times the slope increases and ap-proaches ks + kn. The transition to the final

tion 7’ implies that extrapolating the final slopeto T p = O (shown for top curve) gives the in-

to estimate N VO and kl (see Fig. 9). For all–4 –1solutions k s = 10 s

104 seconds.

104


caused by a number of independent factors, for example, a decrease in the surfacedensity of CD4 or an increase in the time required for the viral envelope to fuse withthe cell membrane and the viral core to enter the cell. A numerical ranking of target-cell tropisms, or affinities, of HIV according to the value of ke would help to clarifywhether reported variations in virulence are due to increased transmission of the virusfrom cell to cell or to increased replication of the virus within a single cell. The con-stant ke is a measure of transmission.

Infection As a Branching Process

In an infectivity assay virions infect target cells (the primary infection phase),then new virions that bud from those infected cells infect other uninfected target cells(secondary infection phase), and so on. Thus an infectivity assay can be likened to abranching process (Fig. 10). Each primary infection generates (on average) Vn sec-ondary virions, which enter the culture medium without pre-incubation (TP = O). The

small number of cells.Blocking secondary infections with sCD4 allows estimation of the branching

number for unblocked infections. Let’s define Bmin as the minimum sCD4 concen-

(1 + BminKassoc). Results from Deen

for Bmin and a value for Kgreater than 300.

That large value may be due to the fact that Deen et al. stimulated the CD4+

lymphocytes in their assay with the mitogen (mitosis-inducing agent) phytohemagglu-tinin (PHA). Recent work by Gowda et al. suggests that stimulation of human CD4+

ESTIMATING kt AND NVO

Fig. 9. Estimation of NVO and ke by using datafrom at least two of the “pre-incubation as-says” simulated In Fig. 8. Extrapolating thefinal slope of the top curve of Fig. 8 to TP = O

1 08 c m–3. Performing a similar extrapolationwhen L = 107 cm-3 (with all other conditions

(graph not shown). Plotting the reciprocal of

straight line with an intercept of 1/NVo and

ks + kn is given by the final slope in Fig. 7, ke

can be estimated directly.

Primary Infection Secondary Infection Tertiary Infection

BRANCHING PROCESSES

Fig. 10. The spread of HIV infection fromcell to cell by free virions can be viewed asa branching process. Here Vn, the expectednumber of progeny virions from an infected

the probability that a progenyvirion will find a target, is 0.25. Since the

process is just self-sustaining.


The Kinetics of HIVInfectivity

Mathematical Considerations

To facilitate analysis of the rate equations governing the kinetics of a viralinfectivity assay (Eqs. 2–5) in the main text), we introduce non-dimensional

(F +C)/NV = f+c. We also introduce non-dimensional time t = (ks + kn)T and the

(l’)

and

(4’)

. . . lead to solutions of Eqs. 1’-4’. The zeroth-order truncation,f = fo and c = co, is equivalent to the usual quasi-steady-state approximation,

Adding Eqs. 3’ and 4’ and applying the steady-state approximation yield

(5’)

lymphocytes by mitogens significantly increases the rate at which they are infectedby HIV. Therefore, it is conceivable that stimulation with PHA significantly increased

Therefore, additional experiments to determine the branching number of both restingand stimulated lymphocytes are needed.

and Bmin

1000 µg cm-3 of sCD4 to treat established infectionsin vivo (a very high concentration). Even more pessimistically, we note that target-cell infection by direct cell-to-cell contact is probably less easily blocked than infec-tion by free virus in the fluid medium. Experiments examining this situation are alsorequired.

The predictions about therapeutic use of sCD4 hold if the only effect of sCD4



which is a form of the Bernoulli equation and can be solved by separation ofvariables.

Substituting g(t) into Eq. 2’ gives an expression for v(t). Next, substituting

approximated by

(6’)

Equation 6’ is related to the incomplete gamma function, and the approximation that

(7’)

of those limits lead to the expressions

and

(9’)

As explained in the main text, Eqs. 7’ and 8’ are useful for the design and

Figure 5 in the main text shows some characteristics of the transition from the

is to block free virus from infecting target cells. Siliciano et al. and Lanzavecchia etal. have suggested that sCD4 may also act to protect CD4+ lymphocytes from indi-rect or autoimmune effects induced by gp 120. If that is the case, then much lowerconcentrations of sCD4 may be of therapeutic use.

The branching number can also be used to estimate the immune response thatan anti -gp 120 vaccine must induce to protect against HIV infection. In that instanceKassoc is the equilibrium constant for association of gp120 and neutralizing antibody(Ig), and Bmin is the minimum concentration of Ig required to extinguish the spreadof infection. Assuming that neutralizing Ig has a Kassoc identical to that of sCD4 (a

0.03 mg cm-3 for blood. For lymph nodes we calculate that aconcentration of about 3 mg cm–3 will be required to prevent growth of infection.Normally, the total concentration of all the hundreds of thousands of antibodies in

extremely high concentration of antibody.



Conclusions

Viral infectivity assays have been an indispensable tool for HIV research. How-ever, we believe that their utility cart be vastly increased by analyzing the kineticprocesses involved rather than treating them like a black box. Analysis of publisheddata using our kinetic model have revealed limitations in present assay designs andambiguities in assay results. For example, blocker assays (see Fig. 7) are usually notdesigned to minimize the error in determining Kassoc nor to answer more than onequestion at a time.

Our immediate goal is to increase the quality and amount of information deriv-able from HIV infectivity assays. We have approached this by defining five parame-

k k., k l, and N Vo characterizing HIV infection of target cells. So farthe model has been used successfully to calculate Kassoc from number of publishedexperiments. That success gives us some initial confidence in our model. We willgain additional confidence by calculating the values of the other parameters from ex-perimental data. Then it may be possible to search for processes not included in themodel by comparing theory and experiment.

We also plan to expand the model to include intracellular processes (for exam-ple, by dividing kl into components describing viral penetration, uncoating, transcrip-tion, maturation, and budding) so that the kinetics of the new class of intracellularblocking agents can be quantified. Finally, our more ambitious goal is to improve theinterpretation of infectivity assays in vitro to the point that reliable extrapolations canbe made to pathogenic processes in vivo.

Our kinetic model is the first attempt to provide a theoretical foundation for theinterpretation of viral infectivity assays, We hope that our presentation makes clearthe practical value of a rigorous mathematical approach to the problem. ■

Further Reading

Keith C. Deen, J. Steven McDougal, Richard Inacker, Gail Folena-Wasserman, Jim Arthos, JonathanRosenberg, Paul Jay Maddon, Richard Axel, and Raymond W. Sweet. 1988. A soluble form of CD4(T4) protein inhibits AIDS virus infection. Nature 331:82-84.

Shantharaj D. Gowda, Barry S. Stein, Nahid Mohagheghpour, Claudia J. Benike, and Edgar G. Engleman.1989. Evidence that T cell activation is required for HIV-1 entry in CD4+ lymphocytes. Journal ofImmunology 142:773–780.

Robert F. Siliciano, Trebor Lawton, Cindy Knall, Robert W. Karr, Phillip Berman, Timothy Gregory, andEllis L. Reinherz. 1988. Analysis of host-virus interactions in AIDS with anti-gp120 T cell clones: Effectof HIV sequence variation and a mechanism for CD4+ cell depletion. Cell 54:561–575.

Antonio Lanzavecchia, Eddy Roosnek, Tim Gregory, Phillip Berman, and Sergio Abrignani. 1988. T cellscan present antigens such as HIV gp120 targeted to their own surface molecules. Nature 334:530-532.

Rebecca E. Hussey, Nneil E. Richardson, Mark Kowalski, Nicholas R. Brown, Hsiu-Ching Chang, RobertF. Siliciano, Tatyana Dorfman, Bruce Walker, Joseph Sodroski, and Eellis L. Reinherz. 1988. A solubleCD4 protein selectively inhibits HIV replication and syncytium formation. Nature 331:78–81.

Cecilia Cheng-Mayer, Deborah Seto, Masatoshi Tateno, and Jay A. Levy. 1988. Biologic features of HIV-1that correlate with virulence in the host. Science 240:8&82.

108

Anthony S. Fauci. 1988. The human immunodeficiency virus: infectivity and mechanisms of pathogenesis.Science 239:617–622.

Los Alamos Science Fall 1989


Amanda G. Fisher, Barbara Ensoli, David Looney, Amdrea Rose, Robert C. Gallo, Michael S. Saag, GeorgeM. Shaw, Beatrice H. Hahn, and Flossie Wong-Staal. 1988. Biologically diverse molecular variants withina single HIV-1 isolate. Nature 334:444-447.

Scott P. Layne, John L. Spouge and Micah Dembo. 1989. Quantifying the infectivity of HIV. Proceedingsof the National Academy of Sciences of the United States of America, in press.

Laurence A. Lasky, Gerald Nakamura, Douglas H. Smith, Christopher Fennie, Craig Shimasaki, Eric Patzer,Phillip Berman, Timothy Gregory, and Daniel J. Capon. 1987. Delineation of a region of the humanimmunodeficiency virus type 1 gp120 glycoprotein critical for interaction with the CD4 receptor. Cell50:975–985.

Peter L. Nara, W. C. Hatch, N. M. Dunlop, W. G. Robey, L. O. Arthur, M. A. Gonda and P. J. Fischinger.1987. Simple, rapid, quantitative, syncytium-forming microassay for the detection of human immunodefi-ciency virus neutralizing antibody. AIDS Research and Human Retroviruses 3:283–302.

M. Ozel, G. Pauli, and H. R. Gelderblom. 1988. The organization of the envelope projections on the surfaceof HIV. Archives of Virology 100:255-266.

Douglas H. Smith, Randal A. Byrn, Scot A. Marsters, Timothy Gregory, Jerome E. Groopman, and DanielJ. Capon. 1987. Blocking of HIV-1 infectivity by a soluble, secreted form of the CD4 antigen. Science238: 1704-1707.

John L. Spouge, Scott P. Layne and Micah Dembo. 1989. Analytic results for quantifying HIV infectivity.Bulletin of Mathematical Biology, in press.

Micah Dembo earned his B.S. in mathematics fromAllegheny College in 1972 and his Ph.D. in biomath-ematics from Cornell University Medical College in1977. After finishing graduate work he came to LosAlamos as a postdoctoral fellow in the TheoreticalBiology and Biophysics Group and remained as astaff member after the appointment ended. Duringhis years in the group, he has worked on a numberof theoretical problems of importance in biology.In addition to developing mathematical models ofcell activation and desensitization, he has workedon the modeling of cooperative interactions in pro-teins, on diffusion reaction problems, particularlywith regard to membrane transport phenomena, andon fluid mechanical models of cell motility. In 1982the National Institutes of Health awarded him a Re-search Career Development Award to work in cellmotility.

Scott P. Layne is a Long Term Visiting Staff Mem-ber in the Mathematical Modeling Group of the The-oretical Division at Los Alamos and a staff mem-ber at Lawrence Livermore National Laboratory.He studied chemistry at Depauw University and at-tended medical school at Case Western Reserve Uni-versity. After finishing an internship he was a post-doctoral fellow at the Center for Nonlinear Studiesat Los Alamos, where he looked for evidence ofnonlinear excitations (solitons) in living cells. Scotthas also studied at Stanford University’s Departmentof Applied Physics. In addition to AIDS epidemi-ology, he works on quantifying the infectivity ofHIV.

John L. Spouge was born in Sheffield, England,and educated at the University of British Columbia,where he earned his B.S. and M.D. degrees and Ox-ford University where he earned a Ph.D. in mathe-matics. He was a postdoctoral fellow at Los Alamosfrom 1983 to 1985. Currently, he is employed at theNational Institutes of Health in Bethesda, Maryland.His research interests include developing systematicstatistical methods of evaluating and organizing se-quence information (such as that found in DNA),examining the physics of diffusion-controlled re-actions, and developing more efficient algorithmsfor sequence alignments with a view to developingprograms that do mutiple-sequence alignments andstructure predictions. This latter work has yielded ageneral theory for improving sequence-alignment al-gorithms as well as code that is demonstrably fasterfor two sequences and that may prove dramaticallyfaster for multiple sequences.


Correction: The photograph in issue Num-ber 17 (National Security Issues) on page34 was misidentified-the flags are flyingfrom a derrick at the U.S. Test Site inNevada rather than at the Soviet's test site inSemipalantinsk. The photographer was DonMacBryde of the Laboratory’s InformationServices.

This report was prepared as an account of worksponsored by the United States Government. Nei-ther the United States Government, nor the UnitedStates Department of Energy, nor any of their em-ployees makes any warranty, expressed or implied,or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of any in-formation, apparatus, product, or process disclosed,or represents that its use would not infringe privatelyowned rights. Reference herein to any specific com-merical product, process, or service by trade name,mark, manufacturer, or otherwise, does not neces-sarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Govern-ment or any agency thereof. The views and opinionsof authors expressed herein do not necessarily stateor reflect those of the United States Government orany agency thereof.

Los Alamos National Laboratory is an Equal Op-portunity Employer conducting a variety of nationaldefense programs and energy-related programs, aswell as a broad spectrum of basic scientific re-search. For scientific and technical employment op-portunities, individuals and academic placement of-ficers are encouraged to contact the Personnel Divi-sion Leader, Los Alamos National Laboratory, LosAlamos, New Mexico, 87545.

LAUR-89-2000

The Kinetics of HIV Infectivity - Federation of American ... · The Kinetics of HIV Infectivity The virologist would set about answering our questions by running a series of viral

Documents